Examples for a problem of locally convex spaces

Hi all!
I'm a math student at the first year of my degree. To begin, sorry for my poor English, it's purely academic. I have a teacher of "Partial differential equations", who gave me exercises named "Exercises of locally convex spaces". All of this without any theory behind it, and I don't really know how to begin with them, since he won't help us and he doesn't reply to our mails. Could anybody give me some hint? Thanks in advance, I'll appreciate any help.

I have to show with explicit examples that the space C(A) of continuous functions in the open set A in R^(n) contains bounded sets that for the compact-open topology are not bounded.

Thank you all :)

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introduction to locally convex topological vector spaces?

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Let E be a normed space, (X,d) a metric space and A βŠ‚ E a convex bounded set. Let f: A β†’ X be uniformly continuous. Show that f(A) is bounded.

I have been struggling with this question for days now and I cannot figure out how to prove it. It doesn't seem to follow the regular epsilon-delta manner. Any help would be appreciated. By convexity of A I have that for any x,y \in A, also kx + (1-k)y \in A and by boundedness of A I have that for any x,y \in A |x-y| <= M.

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What is meant by "locally convex"?

I've checked and can't find a satisfactory explanation online. I see a lot use of the phrase along with "spaces", topological or otherwise. I came face to face with that term in the phrase, "a locally convex curve".

My understanding so far: locally, the "thing" is convex but over the whole, it's not or doesn't have to be. So, no loops and/or the curvature never changes sign? But then, what's that mean? Examples would be helpful!

Thanks.

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my own little proof of the shadow of a convex shape! i did this a few months ago, and i was super excited to see him post a video about it a few weeks ago!
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Any guesses as to why my scoring is concave vs convex?
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Free-roaming horses (alleged wild by the photographer) at Fazenda Barra Mansa in the southern Pantanal. Notice the convex skull and dun coloration, ideal primitive traits. reddit.com/gallery/ryf20x
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Does PC stand for Parcan or Plano-Convex?

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I know we're talking about PARs if they specify MFL, WFL, cp61, cp62, etc. but often they don't. Just a few days ago I got a rider asking for 20 650W PCs. I don't even think anyone supplies 650W PAR lamps in my city. Does that mean they're asking for 650W Plano-Convex spots? I don't even think those exist in my city either. The symbol looks like a fat rectangle with barndoors. Does anyone even use Plano-Convex spots anymore?

Are there any Europeans here that can help me figure out what I'm being asked for? I can't exactly Google "PC" and get a great result...

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It has been conjectured that all 3-dimensional convex polyhedra are Rupert. On the other hand, there is statistical evidence that the rhombicosidodecahedron is probably not Rupert. Thoughts?

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